# Rotational mechanics

by Urmi Roy
Tags: mechanics, rotational
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P: 162
 Quote by Urmi Roy From what I figured, if we have a wheel, on level ground,on which a force is being applied tangentially,this force 'F' serves to accelerate the CM of the wheel aswell as to supply a torque to the wheel. The effect of this torque is felt on all the individual particles of the wheel,of which one is the lowermost point 'P', at which the wheel is in contact with the ground.
'F' would not produce torque at the points of the wheel on its line of action.
For the solid to have a net torque, the torque at its centre of mass must be non-zero.
If 'F' is applied at the centre of mass of the wheel and there is no friction, then the wheel will not rotate. But if 'F' is applied somewhere other than the centre of mass, it will produce a net torque about the centre of mass and set the wheel rotating on its own.
When there is a friction force at 'P', it too produces torque at the centre of mass of the wheel. If 'F' is applied at the centre of mass, the it is the friction force at 'P' that is responsible for setting the wheel in rotational motion.
At least I think so.

 However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),so it moves past its original position,but at uniform velocity(consiering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground). From the perspective of the CM, there is a net force 'F' so CM of the wheel accelerates.
I'm not following you:( How is there no net force on 'P'? If that were so then P would move in a straight line.
I don't understand what you're saying about the movements from different perspectives. Please explain?

 by the way, can we have a force which only has an effective torque,but doesn't cause any linear acceleration of the wheel its working on--as in "an automobile with its engine revved to even 12000 rpm on a frictionless surface, which will stay put with an enormous angular velocity (measured at its wheels) but zero linear velocity."??).I suppose we can find out the angular and linear velocities imparted here separately.
Yes, but if there's friction it will tend to convert some of the angular movement into linear movement . . . the reverse of what it tends to do when the agent provoking movement is a force applied at the cenre of mass instead of a moment.

 Its true that it is difficult to imagine there to be no frictional force for a wheel rotating without slipping,and upon which there is no other force acting,but in this case,I suppose we can say that the 'P' doesn't have any tendency to 'push past' the ground, so in turn, the ground doesn't have to give any reactional force.
I like that way of thinking about it :) 'P' doesn't have any tendency to push past the ground because the wheel has no tendency to accelerate - 'P' pushing against the ground would be to accelerate (or deccelerate) the wheel.
P: 692
 Quote by BobbyBear 'F' would not produce torque at the points of the wheel on its line of action. For the solid to have a net torque,.......At least I think so.
I was just trying to say that the 'F' does effect the state of motion of 'P',and tries to accelerate it.

 Quote by BobbyBear I'm not following you:( How is there no net force on 'P'? If that were so then P would move in a straight line.
Concentrating upon the instant that 'P' is the lowermost point,it does move in a straight line,doesn't it?

 Quote by BobbyBear I don't understand what you're saying about the movements from different perspectives. Please explain?
Well, actually, I found that in rotational mechanics,one tends to say 'torque with respect to CM' or 'torque with respect to any other point',...quite like different frames of references, really.I don't know how appropiate that is in this case,but I just tried it out.

 Quote by BobbyBear Yes, but if there's friction it will tend to convert some of the angular movement into linear movement . . . the reverse of what it tends to do when the agent provoking movement is a force applied at the cenre of mass instead of a moment.
...meaning,basically that this is possible,but friction is not such an example...did I understand that right?

Please bear with me if I'm being a little stupid, but I've always had an ambiguity with this topic.
P: 288
Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of $$\frac{Fm}{M+m}$$ is below the max static friction force)

 However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),
Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :)

 so it moves past its original position,but at uniform velocity(considering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).
No need to consider view points here. Just assume you're in an inertial frame and are provided with all facilities to measure the velocities and accelerations of individual particles as well as the wheel as a whole. Do check out what an inertial frame means if you don't already know. A final year Automobile engineering friend of mine once asked me, while i was explaining some stuff-and simultaneously doubting whether i'm right-, if such a frame called inertial frame really existed or if i'm just bluffing.
Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned, because anything above zero isn't admissible to the earth's surface. Its just like me. tooo lazy!. Watch the cycloid Urmi.
P: 162
 Quote by sganesh88 Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of $$\frac{Fm}{M+m}$$ is below the max static friction force)
Ooooh tricky! that's something I need to ponder over:P:P However, yes, okay . . . but maybe now we need to redefine what 'doing work' is. The friction force would be doing work upon the top block, but it'd merely be transmitting part of the total work (the ratio m/M) beind done by the force F. It'd be acting like an internal force (so long as it's static friction). So maybe what we mean when we say that static friction cannot do work, is that we mean that it does not dissipate energy, thinking that friction in general (kinetic friction) is a dissipative force.
Yes?:P
P: 162
 Quote by sganesh88 A final year Automobile engineering friend of mine once asked me, while i was explaining some stuff-and simultaneously doubting whether i'm right-, if such a frame called inertial frame really existed or if i'm just bluffing.
As far as I've been given to understand, it's Newton's first law that claims the existence of inertial frames.
P: 288
 Yes?:P
I'm afraid, no. Why are you bothered about static friction doing work anyway? Btw i was also irked when i heard gravity does work.
P: 288
 As far as I've been given to understand, it's Newton's first law that claims the existence of inertial frames.
Yes. So anyway first law created it. I didn't.
P: 162
About the net force on 'P' (I'm assuming we are considering the situation that Urmi Roy described: the wheel subject to a tangential force 'F' (applied at its centre of mass?), and a static friction force at the point of contact 'P', thus the wheel has both a linear acceration and an angular acceleration (both of them compatible with the no slipping condition).

 Quote by Urmi Roy However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),so it moves past its original position,but at uniform velocity(consiering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).
 Quote by sganesh88 Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :) [...] Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned, because anything above zero isn't admissible to the earth's surface. Its just like me. tooo lazy!. Watch the cycloid Urmi.
I'm not sure how you'd know what forces are acting on individual particles of the wheel. Are you working it out from the movement you know of the wheel, from which you know the movement of each particle of the wheel, as we're considering the solid to be rigid?
So! basically, the centre of mass has a linear acceleration (and no angular acceleration), so the net force upon it is a linear force to the right.
And all points, including point 'P', have the same linear acceleration as the centre of mass, plus an angular acceleration about the centre of mass (superposition of two movemets). Thus, they are all subject to the same force that the centre of mass is, plus a centripetal force directed toward the centre of mass equal to the 'mass of the particle' times the distance of the particle to the centre of mass and the square of the angular velocity of the wheel at each instant.
P: 162
 Quote by sganesh88 I'm afraid, no. Why are you bothered about static friction doing work anyway? Btw i was also irked when i heard gravity does work.
Oh :(
but I'm correct in saying that in your example the friction force doesn't do 'it's own' work, just transmits part of the work done by the force 'F'. And its true that friction cannot provoke movement on its own, it cannot transform some other kind of energy into kinetic energy! that's all I'm trying to say. At least this is true?
Mentor
P: 41,440
 Quote by sganesh88 Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of $$\frac{Fm}{M+m}$$ is below the max static friction force)
Good point. Whether a force does work on a system depends on the reference frame used to analyze the system. But the key point about static friction is that it's a passive force. For the "real" source of the energy used to accelerate the top block one must look to what's doing the work on the lower block.
P: 692
 Quote by sganesh88 Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface.
Thanks,this really helped to clear my concepts further!!

 Quote by sganesh88 Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :)
Will this have a serious bearing on the point of view I have formed about this entire event of roling without slipping?

 Quote by sganesh88 No need to consider view points here. Just assume you're in an inertial frame and are provided with all facilities to measure the velocities and accelerations of individual particles as well as the wheel as a whole. Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned.
Just very cautiously,let me ask if this idea of considering view points,or rather frames of reference is actually wrong,even if I don't need it here.

Also, please tell me how I could modify my understanding of 'rolling' by stating what is wrong and what is right about my post(post 72 of this thread).
P: 162
 Quote by Urmi Roy Well, actually, I found that in rotational mechanics,one tends to say 'torque with respect to CM' or 'torque with respect to any other point',...quite like different frames of references, really.I don't know how appropiate that is in this case,but I just tried it out.
The moment produced by a force is not uniform in space, it depends upon the point you consider. Only the moment produced by a pair of forces of equal magnitude and opposite direction whose lines of action do not coincide produce a uniform moment in all of space.
That's why we talk of the moment (or torque) with respect to a point.

Maybe you're mixing this idea with different referece frames? I really don't think there's any relationship, but if you find there is let me know!
P: 288
 I'm not sure how you'd know what forces are acting on individual particles of the wheel. Are you working it out from the movement you know of the wheel, from which you know the movement of each particle of the wheel, as we're considering the solid to be rigid?
Assuming pure rolling;taking the mass of the particle to be some $$\Delta$$m, we can compute the instantanoues force acting on it at a particular position w.r.t COM.

 Thus, they are all subject to the same force that the centre of mass is, plus a centripetal force directed toward the centre of mass equal to the 'mass of the particle' times the distance of the particle to the centre of mass and the square of the angular velocity of the wheel at each instant.
Considering pure rolling, the contact point P should have a zero horizontal velocity component, whatever the tangential force. The only unbalanced force acting on it is the centripetal force.

 but I'm correct in saying that in your example the friction force doesn't do 'it's own' work, just transmits part of the work done by the force 'F'. And its true that friction cannot provoke movement on its own, it cannot transform some other kind of energy into kinetic energy! that's all I'm trying to say.
Going by the definition of work done which is, the dot product of the force and displacement, static friction can claim that it can do work. But yes, it can't cause motion on its own; i.e., without another force entering the picture.
P: 288
 Quote by Urmi Roy Will this have a serious bearing on the point of view I have formed about this entire event of roling without slipping?
Rolling can be analysed without bothering about what happens at the "particle" level by just considering the wheel as a rigid body and computing the external forces and torques(w.r.t COM preferably) on it. But understanding the complex motions of particles is also quite fun.

 Just very cautiously,let me ask if this idea of considering view points,or rather frames of reference is actually wrong,even if I don't need it here. Also, please tell me how I could modify my understanding of 'rolling' by stating what is wrong and what is right about my post(post 72 of this thread).
Reference frames are important when you describe motion. And you've come a long way from post #1. Much faster than me, i confess. Just keep thinking about it. Refer some books and internet articles. Anyway is this just for understanding purpose or are you giving some lecture on it?
P: 692
 Quote by sganesh88 Rolling can be analysed without bothering about what happens at the "particle" level ...... But understanding the complex motions of particles is also quite fun.
I thought that perhaps analysing in this way might help in my understanding,but if you think it's not, I'll try in a different way.

 Quote by sganesh88 Reference frames are important when you describe motion. And you've come a long way from post #1. Much faster than me, i confess.
BobbyBear said that the usage of 'frames of reference' in rotational mechanics isn't quite correct.Please tell me where I'm going wrong.

 Quote by sganesh88 Just keep thinking about it. Refer some books and internet articles. Anyway is this just for understanding purpose or are you giving some lecture on it?
Pleeease don't leave it at that! I don't want to just improve in this,I want to finally remove this doubt that has been bothering me for so long,even though its just for the sake of my satisfaction,I don't have any lectures to give! I can only do it if you all help me!
P: 288
 Quote by Urmi Roy I thought that perhaps analysing in this way might help in my understanding,but if you think it's not, I'll try in a different way.
Yes. it does. But to understand the motion of the wheel as a whole, the former approach would suffice.

 BobbyBear said that the usage of 'frames of reference' in rotational mechanics isn't quite correct.Please tell me where I'm going wrong.
BobbyBear said reference frames aren't analogous to reference points w.r.t which moments are measured. Quantities like displacement,force, velocity and the whole lot of physical quantities lose their meaning in the absence of a reference frame. My initial doubt has been confirmed. Read about reference frames. Physicsforums and wikipedia would help you in it, i'm sure.
P: 288
 Quote by Urmi Roy Pleeease don't leave it at that! I don't want to just improve in this,I want to finally remove this doubt that has been bothering me for so long,even though its just for the sake of my satisfaction,I don't have any lectures to give! I can only do it if you all help me!
What is the doubt that you presently have?
 P: 692 Actually I don't have anything new. As I said,this issue of 'rolling friction' has been bothering me for quite a while,the main difficulty of which stemmed from a particular extract of a book I read. So,after getting together all that Doc Al, BobbyBear, vin300 and ofcourse, you said, I just tried to summarise the new picture in my head, in post 72. Since then,you pointed out that viewing it from the aspect of individual particles isn't necessary, so apart from that, I just wanted to confirm my idea (as presented in #72) was basically right. If you think I have any major problems, please modify my post and tell me the true 'picture'.

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